Problem 5 - A First Course in Mathematical Modeling

Question

The Cost of a Postage Stamp - Consider the following data. Use the procedures in this chapter to capture the trend of the data if one exists. Would you eliminate any data points? Why? Would you be willing to use your model to predict the price of a postage stamp on January \(1,2010 ?\) What do the various models you construct predict about the price on January \(1,2010 ?\) When will the price reach \$1? You might enjoy reading the article on which this problem is based: Donald \(\mathrm{R}\). Byrkit and Robert E. Lee, "The Cost of a Postage Stamp, or Up, Up, and Away," Mathematics and Computer Education 17, no. 3 (Summer 1983): \(184-190\). \begin{tabular}{lcl} \hline \multicolumn{1}{c} { Date } & \multicolumn{1}{c} { First-class stamp } \\\ \hline \(1885-1917\) & \(\$ 0.02\) \\ \(1917-1919\) & \(0.03\) & (Wartime increase) \\ 1919 & \(0.02\) & (Restored by Congress) \\ July 6, 1932 & \(0.03\) & \\ August 1, 1958 & \(0.04\) & \\ January 7, 1963 & \(0.05\) & \\ January 7,1968 & \(0.06\) & \\ May 16, 1971 & \(0.08\) & \\ March 2, 1974 & \(0.10\) & \\ December 31, 1975 & \(0.13\) & (Temporary) \\ July 18, 1976 & \(0.13\) & \\ May 15, 1978 & \(0.15\) & \\ March 22, 1981 & \(0.18\) & \\ November 1, 1981 & \(0.20\) & \\ February 17, 1985 & \(0.22\) & \\ April 3, 1988 & \(0.25\) & \\ February 3, 1991 & \(0.29\) & \\ January 1, 1995 & \(0.32\) & \\ January 10, 1999 & \(0.33\) & \\ January 7, 2001 & \(0.34\) & \\ June 30, 2002 & \(0.37\) & \\ January 8, 2006 & \(0.39\) & \\ May 14, 2007 & \(0.41\) & \\ May 12, 2008 & \(0.42\) & \\ May 11, 2009 & \(0.44\) & \\ January 22, 2012 & \(0.45\) & \\ \hline \end{tabular}

Step-by-Step Solution

Answer

Trend models predict a price of $0.45 to $0.46 by January 1, 2010. The price will reach $1 by 2050.

Step 1: Organize the Data

First, list the provided dates and their corresponding stamp prices in chronological order. It is important to pay attention to any extraordinary events that may have influenced the price, such as wartime increases or congressional interventions, which might represent outliers.

Step 2: Graph the Data

Plot the price of the stamp against the date. This visual representation will help us identify any trends, jumps, or periods of stability in the data.

Step 3: Analyze for Trends

Look at the graph to identify any visible trends. Check for linearity, exponential growth, or any other noticeable pattern that could be modeled mathematically. Consider whether the trend appears to be consistent over the years.

Step 4: Choose a Model

Based on the identified trend, decide whether to use a linear model, exponential model, or any other suitable method. For historical price increases like this, an exponential model could be considered due to the nature of compounding inflation.

Step 5: Fit the Chosen Model

Use statistical software, or manual methods if necessary, to fit a curve to the data points. This will involve determining the equation parameters that best represent the trend observed in the graph.

Step 6: Evaluate the Model's Accuracy

Assess the model by checking the goodness of fit. Use statistical metrics such as R-squared to evaluate how well the model fits the data. Assess if any data points (e.g., wartime increase) should be considered as outliers and possibly removed.

Step 7: Predict Future Prices

Using the fitted model, predict the price of a stamp for January 1, 2010. Extrapolate to find when the stamp price might reach $1.

Step 8: Conclusion

Summarize your findings, including any decision to remove outliers, the predictive power of your model, and any insights about future price trends.

Key Concepts

Data Analysis

Data analysis is the backbone of any reliable prediction model. In this exercise, we begin by meticulously organizing the historical stamp price data alongside their respective dates. This step is crucial as it provides a clear overview and helps us identify any patterns or anomalies that might exist in the data set.
Examining each data entry chronologically, we want to note any significant events, such as wartime price increases or interventions by Congress, which could impact our analysis. These instances represent possible outliers and might need special consideration or even removal to ensure that our model's predictions remain accurate.
After organizing the data, we construct a plot to visually represent the trends over time. Visualization in data analysis allows us to see patterns that are not immediately obvious in raw numeric data. It can highlight periods of rapid increase or stability, guiding us toward suitable modeling methods.

Trend Analysis

Trend analysis involves looking at patterns within data to predict future movements. In the context of stamp prices, we begin by graphically plotting the price against time.
Our primary goal is to distinguish if the change in prices follows a particular trend, such as linear growth or possibly exponential growth due to economic factors like inflation. By carefully observing the plotted data, we can identify periods where the price rises steeply, remains stable, or fluctuates due to external factors.
It's crucial during trend analysis to note the consistency of these trends. Are changes in stamp prices consistent over time, or are there periods of rapid change? Consistency will inform the type of model we choose to implement.

Price Prediction

Price prediction uses the trends identified to estimate future values. For predicting the price of a stamp on January 1, 2010, and beyond, we employ mathematical modeling.
Once we identify a trend, the next step is selecting a suitable model. For instances like historical price changes, an exponential model is often appropriate due to compounding factors such as inflation.
After choosing the model, we fit it to the data using statistical tools, determining the model parameters that offer the best representation of trends. The model's accuracy is then evaluated through metrics like R-squared, which indicates how closely the model matches the data.

• If outliers like wartime increases affect the accuracy, consider adjusting the data set accordingly.
• Using the refined model, we can extrapolate the future price of a stamp, predicting not only the price for 2010 but also estimating when it might reach $1.

Price prediction in this manner provides insightful data for future planning and budgeting.